General Index Accelerated convergence of series 166ff. schedule 445 Accuracy 28f. thermodynamic analogy 444f. achievable in minimization 398,404,410 traveling salesman problem 445ff. achievable in root finding 353 ANSI C standard 2f.,14,25,930,941 contrasted with fidelity 841,849 ANSI macro 17,930 CPU different from memory 186 Antonov-Saleev variant of Sobol'sequence vs.stability 710,736,839,853 310E Acknowledgments xii Apple xvii Adams-Bashford-Moulton method 749 Macintosh 894 Adams'stopping criterion 373 Approximate inverse of matrix 57 Adaptive integration 129,141.709.714ff.. Approximation of functions 105f. 725ff,733f,737,744,749f,797 by Chebyshev polynomials 191f,519 Adaptive Monte Carlo integration 316ff., Pade approximant 200ff. 319f by rational functions 204ff Addition,multiple precision 916 by wavelets 601f..791 Addition theorem,elliptic integrals 262 see also Fitting ADI (altemating direction implicit)method Arguments,conversion of data types 24f.,930 856.870f.915 Arithmetic Adjoint operator 876 arbitrary precision 889.915ff. Adobe Illustrator xiii,xvii complex 23f.,948ff. Advective equation 835 floating point 889 AGM (arithmetic geometric mean)915 IEEE standard 285,890f Airy function 210,240.250 rounding 890 routine for 250f. Arithmetic coding 889,910ff. Aitken's delta squared process 166 Arithmetic-geometric mean (AGM)method Aitken's interpolation algorithm 108 915 Algorithms,non-numerical 889ff. Array Aliasing 501,576 centered subarray of 119 see also Fourier transform how to allocate 19 All-poles model 573 index range 18 see also Maximum entropy method(MEM) one-dimensional 18 All-zeros model 573 relation to C pointer 18 see also Periodogram three-dimensional 23 Allocation of storage 19,21f.,940ff. two-dimensional 20f. Alternating-direction implicit method (ADI) unit-offset 18,940f. 856.870f,915 variable dimension 20 Alternating series 166f zero-offset 18 Alternative extended Simpson's rule 134 Artificial viscosity 840,846 Amoeba 410 Ascending transformation,elliptic integrals see also Simplex,method of Nelder and 262 Mead ASCII character set 5,896,903,910 Amplification factor 837.839.841,849,854f Assembly language 278 Amplitude error 840 Associated Legendre polynomials 252f.,773 Analog-to-digital converter 821,894 recurrence relation for 253 Analyticity 201 relation to Legendre polynomials 252 Analyze/factorize/operate package 71f.,833 Association,measures of 610,628ff. Anderson-Darling statistic 626f. Asymptotic series 167 Andrew's sine 702 exponential integral 224 Annealing,method of simulated 394f.,444ff. Attenuation factors 590 assessment 454f. Autocorrelation for continuous variables 444,451f. in linear prediction 565 965
General Index Accelerated convergence of series 166ff. Accuracy 28f. achievable in minimization 398, 404, 410 achievable in root finding 353 contrasted with fidelity 841, 849 CPU different from memory 186 vs. stability 710, 736, 839, 853 Acknowledgments xii Adams-Bashford-Moulton method 749 Adams’ stopping criterion 373 Adaptive integration 129, 141, 709, 714ff., 725ff., 733f., 737, 744, 749f., 797 Adaptive Monte Carlo integration 316ff., 319ff. Addition, multiple precision 916 Addition theorem, elliptic integrals 262 ADI (alternating direction implicit) method 856, 870f., 915 Adjoint operator 876 Adobe Illustrator xiii, xvii Advective equation 835 AGM (arithmetic geometric mean) 915 Airy function 210, 240, 250 routine for 250f. Aitken’s delta squared process 166 Aitken’s interpolation algorithm 108 Algorithms, non-numerical 889ff. Aliasing 501, 576 see also Fourier transform All-poles model 573 see also Maximum entropy method (MEM) All-zeros model 573 see also Periodogram Allocation of storage 19, 21f., 940ff. Alternating-direction implicit method (ADI) 856, 870f., 915 Alternating series 166f. Alternative extended Simpson’s rule 134 Amoeba 410 see also Simplex, method of Nelder and Mead Amplification factor 837, 839, 841, 849, 854f. Amplitude error 840 Analog-to-digital converter 821, 894 Analyticity 201 Analyze/factorize/operate package 71f., 833 Anderson-Darling statistic 626f. Andrew’s sine 702 Annealing, method of simulated 394f., 444ff. assessment 454f. for continuous variables 444, 451f. schedule 445 thermodynamic analogy 444f. traveling salesman problem 445ff. ANSI C standard 2f., 14, 25, 930, 941 ANSI macro 17, 930 Antonov-Saleev variant of Sobol’ sequence 310ff. Apple xvii Macintosh 894 Approximate inverse of matrix 57 Approximation of functions 105f. by Chebyshev polynomials 191f., 519 Pade approximant 200ff. ´ by rational functions 204ff. by wavelets 601f., 791 see also Fitting Arguments, conversion of data types 24f., 930 Arithmetic arbitrary precision 889, 915ff. complex 23f., 948ff. floating point 889 IEEE standard 285, 890f. rounding 890 Arithmetic coding 889, 910ff. Arithmetic-geometric mean (AGM) method 915 Array centered subarray of 119 how to allocate 19 index range 18 one-dimensional 18 relation to C pointer 18 three-dimensional 23 two-dimensional 20f. unit-offset 18, 940f. variable dimension 20 zero-offset 18 Artificial viscosity 840, 846 Ascending transformation, elliptic integrals 262 ASCII character set 5, 896, 903, 910 Assembly language 278 Associated Legendre polynomials 252f., 773 recurrence relation for 253 relation to Legendre polynomials 252 Association, measures of 610, 628ff. Asymptotic series 167 exponential integral 224 Attenuation factors 590 Autocorrelation in linear prediction 565 965
966 Index use of FFT 545 reflection formulas,modified functions Wiener-Khinchin theorem 498,574 247 AUTODIN-II polynomial 898 routines for 232ff.,243ff Autonomous differential equations 735f. routines for modified functions 248f. Autoregressive model (AR)see Maximum en- series for 166,230 tropy method(MEM) series for K 247 Average deviation of distribution 611 series for Y,242 Averaging kernel,in Backus-Gilbert method spherical 240,251 816 turning point 241 Wronskian 240.246 Best-fit parameters 656,662,666,703 Backsubstitution 42,47,50,98 see also Fitting in band diagonal matrix 54 Beta function 213 in Cholesky decomposition 97 incomplete see Incomplete beta function complex equations 49 BFGS algorithm see Broyden-Fletcher-Goldfarb- direct for computing A-1.B 48 Shanno algorithm relaxation solution of boundary value prob- Bias,of exponent 28 lems 764 Bias,removal in linear prediction 570 in singular value decomposition 64 Biconjugacy 84 Backtracking 427 Biconjugate gradient method in quasi-Newton methods 384 elliptic partial differential equations 833 Backus-Gilbert method 815ff preconditioning 85f.,833 Backward deflation 370 for sparse system 84f.,606 Bader-Deufthard method 737.742f. Bicubic interpolation 125f. Bairstow's method 371,376f. Bicubic spline 127f. Balancing 483 Band diagonal matrix 50,51ff Big-endian 302 Bilinear interpolation 123f backsubstitution 54 LU decomposition 53f. Binomial coefficients 213 multiply by vector 52f recurrences for 215 storage 52 Binomial probability function 215 Band-pass filter 558,562 cumulative 229 wavelets 592,599f. deviates from 290,295f. Bandwidth limited function 501 Binormal distribution 637,695 Bank accounts,checksum for 902 Biorthogonality 84 Bar codes,checksum for 902 Bisection 117,366 Bartlett window 554 compared to minimum bracketing 397f., 300f Base of representation 28,890 BASIC,Numerical Recipes in xv,1 minimum finding with derivatives 406 Basis functions in general linear least squares root finding350.353f.,359f.397.476 671 BISYNCH 898 Bayes'Theorem 819 Bit 28 Bayesian reversal in fast Fourier transform(FFT) approach to inverse problems 808,820, 505f,532 825f Bitwise logical functions 296ff.,898f. contrasted with frequentist 819 Block-by-block method 797 vs.historic maximum entropy method Block of statements 6 825f Bode's rule 132 views on straight line fitting 670 Boltzmann probability distribution 445 Bays'shuffle 280 Boltzmann's constant 445 Beroulli number 138 Bootstrap method 691f. Bessel functions 230ff.,240ff. Bordering method for Toeplitz matrix 92f. asymptotic form 230,236 Borwein and Borwein method for 915 complex 210 Boundary 161f..432f..753 continued fraction 240f,246f Boundary conditions double precision 230 for differential equations 707f. fractional order 230,240ff. initial value problems 708 Miller's algorithm 181,234 in multigrid method 877f. modified 236ff. partial differential equations 514,828ff., modified,fractional order 246ff. 857f modified,normalization formula 239,246 for spheroidal harmonics 774 modified,routines for 237ff two-point boundary value problems 708, normalization formula 181 753狂 recurrence relation 178,231,239,241f. Boundary value problems see Differential reflection formulas 242 equations;Elliptic partial differential
966 Index use of FFT 545 Wiener-Khinchin theorem 498, 574 AUTODIN-II polynomial 898 Autonomous differential equations 735f. Autoregressive model (AR) see Maximum entropy method (MEM) Average deviation of distribution 611 Averaging kernel, in Backus-Gilbert method 816 Backsubstitution 42, 47, 50, 98 in band diagonal matrix 54 in Cholesky decomposition 97 complex equations 49 direct for computing A−1 · B 48 relaxation solution of boundary value problems 764 in singular value decomposition 64 Backtracking 427 in quasi-Newton methods 384 Backus-Gilbert method 815ff. Backward deflation 370 Bader-Deuflhard method 737, 742f. Bairstow’s method 371, 376f. Balancing 483 Band diagonal matrix 50, 51ff. backsubstitution 54 LU decomposition 53f. multiply by vector 52f. storage 52 Band-pass filter 558, 562 wavelets 592, 599f. Bandwidth limited function 501 Bank accounts, checksum for 902 Bar codes, checksum for 902 Bartlett window 554 Base of representation 28, 890 BASIC, Numerical Recipes in xv, 1 Basis functions in general linear least squares 671 Bayes’ Theorem 819 Bayesian approach to inverse problems 808, 820, 825f. contrasted with frequentist 819 vs. historic maximum entropy method 825f. views on straight line fitting 670 Bays’ shuffle 280 Bernoulli number 138 Bessel functions 230ff., 240ff. asymptotic form 230, 236 complex 210 continued fraction 240f., 246f. double precision 230 fractional order 230, 240ff. Miller’s algorithm 181, 234 modified 236ff. modified, fractional order 246ff. modified, normalization formula 239, 246 modified, routines for 237ff. normalization formula 181 recurrence relation 178, 231, 239, 241f. reflection formulas 242 reflection formulas, modified functions 247 routines for 232ff., 243ff. routines for modified functions 248f. series for 166, 230 series for Kν 247 series for Yν 242 spherical 240, 251 turning point 241 Wronskian 240, 246 Best-fit parameters 656, 662, 666, 703 see also Fitting Beta function 213 incomplete see Incomplete beta function BFGS algorithm see Broyden-Fletcher-GoldfarbShanno algorithm Bias, of exponent 28 Bias, removal in linear prediction 570 Biconjugacy 84 Biconjugate gradient method elliptic partial differential equations 833 preconditioning 85f., 833 for sparse system 84f., 606 Bicubic interpolation 125f. Bicubic spline 127f. Big-endian 302 Bilinear interpolation 123f. Binomial coefficients 213 recurrences for 215 Binomial probability function 215 cumulative 229 deviates from 290, 295f. Binormal distribution 637, 695 Biorthogonality 84 Bisection 117, 366 compared to minimum bracketing 397f., 399f. minimum finding with derivatives 406 root finding 350, 353f., 359ff., 397, 476 BISYNCH 898 Bit 28 reversal in fast Fourier transform (FFT) 505f., 532 Bitwise logical functions 296ff., 898f. Block-by-block method 797 Block of statements 6 Bode’s rule 132 Boltzmann probability distribution 445 Boltzmann’s constant 445 Bootstrap method 691f. Bordering method for Toeplitz matrix 92f. Borwein and Borwein method for π 915 Boundary 161f., 432f., 753 Boundary conditions for differential equations 707f. initial value problems 708 in multigrid method 877f. partial differential equations 514, 828ff., 857ff. for spheroidal harmonics 774 two-point boundary value problems 708, 753ff. Boundary value problems see Differential equations; Elliptic partial differential
Index 967 equations;Two-point boundary value CCITT (Comite Consultatif International Tele- problems graphique et Telephonique)897f.909 Box-Muller algorithm for normal deviate 289 CCITT polynomial 897f. Bracketing Center of mass 305ff. of function minimum 350,397ff.,409 Central limit theorem 658f. of roots348.350f,360.369,371,376 Central tendency,measures of 610ff. 397 Change of variable Branch cut,for hypergeometric function 209f. in integration 144ff.,797 Branching 8 in Monte Carlo integration 307f. Break iteration 12f. in probability distribution 287ff. Brenner,N.M.506,522 Characteristic polynomial Brent's method digital filter 561 minimization 395f.,402ff.,666 minimization,using derivative 396,406 eigensystems 456,475f. linear prediction 567 root finding 348,356,666 matrix with a specified 375 Broyden-Fletcher-Goldfarb-Shanno algorithm of recurrence relation 180 397,426ft. Broyden's method 380,389ff.,393 Characteristics of partial differential equations 827 singular Jacobian 393 Chebyshev acceleration in successive over- Bubble sort 330 relaxation (SOR)868f. Bugs in compilers xiii Chebyshev approximation 91,130.189,190ff. Clenshaw-Curtis quadrature 196 how to report iv,xviii Clenshaw's recurrence formula 193 Bulirsch-Stoer coefficients for 191 algorithm for rational function interpolation 111f. contrasted with Pade approximation 201 method (differential equations)209.272. derivative of approximated function 189. 708f,712,722,724f,733,747 195 method (differential equations),stepsize economization of series 198ff.,201 control 725,733f. for error function 220f. for second order equations 733 even function 194 Burg's LP algorithm 568 and fast cosine transform 519 Byte 28 gamma functions 242 integral of approximated function 195 odd function 194 C+7,24 polynomial fits derived from 197 C(programming language)11 rational function 204ff. ANSI2f,14,25,930,941 Remes exchange algorithm for filter 560 C++7.24 Chebyshev polynomials 190ff. compilers 3 continuous orthonormality 190f. discrete orthonormality 191 control structures 5 deficiencies 16,24f.,26f. explicit formulas for 190 exteral functions 25 formula for x in terms of 199 Check digit 901 features 15f. function declaration 17 Checksum 889,896 function definition 17 cyclic redundancy (CRC)896ff. header (.h)file 17 Cherry,sundae without a 818 implicit conversions 24f,930 Chi-by-eye 657 Kernighan and Ritchie 2,16,24,930 Chi-square fitting see Fitting;Least squares nature of 15f. fitting Numerical Recipes in xv,I Chi-square probability function 216,221, operator associativity 25f. 621,660,806 operator precedence 25f. as boundary of confidence region 693f. prototypes 2,25,930 related to incomplete gamma function 221 vectors in 18 Chi-square test 620f. Calendar algorithms If.,11ff. for binned data 620f. Calibration 659 chi-by-eye 657 Cards,sorting a hand of 330 and confidence limit estimation 693f. Carlson's elliptic integrals 261f. for contingency table 630ff. Cash-Karp parameters 716f. degrees of freedom 621f. Cauchy probability distribution see Lorentzian for inverse problems 806 probability distribution least squares fitting 659ff. Cauchy problem for partial differential equa- nonlinear models 681ff. tions 827f. rule of thumb 661 Cayley's representation of exp(-iHt)853 for straight line fitting 661ff
Index 967 equations; Two-point boundary value problems Box-Muller algorithm for normal deviate 289 Bracketing of function minimum 350, 397ff., 409 of roots 348, 350ff., 360, 369, 371, 376, 397 Branch cut, for hypergeometric function 209f. Branching 8 Break iteration 12f. Brenner, N.M. 506, 522 Brent’s method minimization 395f., 402ff., 666 minimization, using derivative 396, 406 root finding 348, 356, 666 Broyden-Fletcher-Goldfarb-Shanno algorithm 397, 426ff. Broyden’s method 380, 389ff., 393 singular Jacobian 393 Bubble sort 330 Bugs in compilers xiii how to report iv, xviii Bulirsch-Stoer algorithm for rational function interpolation 111f. method (differential equations) 209, 272, 708f., 712, 722, 724ff., 733, 747 method (differential equations), stepsize control 725, 733f. for second order equations 733 Burg’s LP algorithm 568 Byte 28 C++ 7, 24 C (programming language) 11 ANSI 2f., 14, 25, 930, 941 C++ 7, 24 compilers 3 control structures 5 deficiencies 16, 24f., 26f. external functions 25 features 15f. function declaration 17 function definition 17 header (.h) file 17 implicit conversions 24f., 930 Kernighan and Ritchie 2, 16, 24, 930 nature of 15f. Numerical Recipes in xv, 1 operator associativity 25f. operator precedence 25f. prototypes 2, 25, 930 vectors in 18 Calendar algorithms 1f., 11ff. Calibration 659 Cards, sorting a hand of 330 Carlson’s elliptic integrals 261f. Cash-Karp parameters 716f. Cauchy probability distribution see Lorentzian probability distribution Cauchy problem for partial differential equations 827f. Cayley’s representation of exp(−iHt) 853 CCITT (Comite Consultatif International T ´ el´ e-´ graphique et Tel´ ephonique) 897f., 909 ´ CCITT polynomial 897f. Center of mass 305ff. Central limit theorem 658f. Central tendency, measures of 610ff. Change of variable in integration 144ff., 797 in Monte Carlo integration 307f. in probability distribution 287ff. Characteristic polynomial digital filter 561 eigensystems 456, 475f. linear prediction 567 matrix with a specified 375 of recurrence relation 180 Characteristics of partial differential equations 827 Chebyshev acceleration in successive overrelaxation (SOR) 868f. Chebyshev approximation 91, 130, 189, 190ff. Clenshaw-Curtis quadrature 196 Clenshaw’s recurrence formula 193 coefficients for 191 contrasted with Pade approximation 201 ´ derivative of approximated function 189, 195 economization of series 198ff., 201 for error function 220f. even function 194 and fast cosine transform 519 gamma functions 242 integral of approximated function 195 odd function 194 polynomial fits derived from 197 rational function 204ff. Remes exchange algorithm for filter 560 Chebyshev polynomials 190ff. continuous orthonormality 190f. discrete orthonormality 191 explicit formulas for 190 formula for xk in terms of 199 Check digit 901 Checksum 889, 896 cyclic redundancy (CRC) 896ff. Cherry, sundae without a 818 Chi-by-eye 657 Chi-square fitting see Fitting; Least squares fitting Chi-square probability function 216, 221, 621, 660, 806 as boundary of confidence region 693f. related to incomplete gamma function 221 Chi-square test 620f. for binned data 620f. chi-by-eye 657 and confidence limit estimation 693f. for contingency table 630ff. degrees of freedom 621f. for inverse problems 806 least squares fitting 659ff. nonlinear models 681ff. rule of thumb 661 for straight line fitting 661ff
968 Index for straight line fitting,errors in both coor- Complex systems of linear equations 49f. dinates 666 complex.c utility functions 23f..948ff. for two binned data sets 622 Compression of data 603,889,903ff.,910ff unequal size samples 623 Concordant pair for Kendall's tau 642f. Chip rate 300 Condition number 61,85 Chirp signal 563 Confidence level 692f.,696ff. Cholesky decomposition 96ff.,430.462 Confidence limits backsubstitution 97 bootstrap method 692f operation count 97 and chi-square 693f. pivoting 97 confidence region,confidence interval 692 solution of normal equations 674 on estimated model parameters 689ff. Circulant 592 by Monte Carlo simulation 689ff. Class,data type 7 from singular value decomposition (SVD) Clenshaw-Curtis quadrature 130,196,518 698 519 Confluent hypergeometric function 210,246 Clenshaw's recurrence formula 181ff.,196 Conjugate directions 414f.,421ff. for Chebyshev polynomials 193 Conjugate gradient method stability 181ff. biconjugate 84 Clocking errors 899 compared to variable metric method 425f. cn function 269 elliptic partial differential equations 833 Coarse-to-fine operator 873 for minimization 396f.,420ff.,812f,824 Coarse-grid correction 873f. minimum residual method 85 Coding preconditioner 85f. arithmetic 910ff. for sparse system 83ff.,606 checksums 896 and wavelets 606 decoding a Huffman-encoded message Conservative differential equations 732f 905 Constrained linear inversion method 808ff. Huffman 903ff. Constrained linear optimization see Linear pro- run-length 909 gramming variable length code 903 Constrained optimization 394 Ziv-Lempel 903 Constraints,deterministic 813ff. see also Arithmetic coding;Huffman cod- Constraints,linear 431 ing Contingency coefficient C 631 Coefficients Contingency table 628ff.,644 binomial 215 statistics based on chi-square 630f. for Gaussian quadrature 147ff. statistics based on entropy 632ff. for Gaussian quadrature,nonclassical weight continue construction 14 function 157ff.,797 Continued fraction 169ff. for quadrature formulas 131f,797 Bessel functions 240f. Column degeneracy 32 convergence criterion 171 Column operations on matrix 37,40f. equivalence transformation 172 Column totals 630 evaluation 169ff. Combinatorial minimization see Annealing evaluation along with normalization condi- Comite Consultatif Intemational Telegraphique tion 247 et Telephonique (CCITT)897f.,909 even and odd parts 172,217,222 Communication theory,use in adaptive integra- even part 255,257 tion 727 exponential integral 222 Communications protocol 896 Fresnel integral 255 Comparison function for rejection method incomplete beta function 227 290f incomplete gamma function 217 Complementary error function Lentz's method 171,219 see Error function modified Lentz's method 171 Complete elliptic integral see Elliptic integrals Pincherle's theorem 181 Complex arithmetic 23f.,176ff.,948ff. ratio of Bessel functions 246 avoidance of in path integration 209 rational function approximation 170,217, cubic equations 185 227 for linear equations 49f. recurrence for evaluating 170f. quadratic equations 184 and recurrence relation 181 Complex error function 259 sine and cosine integrals 257 Complex plane Steed's method 170f. fractal structure for Newton's rule 367f. tangent function 169 path integration for function evaluation typography for 169 208ff,271 Continuous variable (statistics)628 poles in11l,166,208f,213,561,573, Control structures 6,8ff. 724f bad 14
968 Index for straight line fitting, errors in both coordinates 666 for two binned data sets 622 unequal size samples 623 Chip rate 300 Chirp signal 563 Cholesky decomposition 96ff., 430, 462 backsubstitution 97 operation count 97 pivoting 97 solution of normal equations 674 Circulant 592 Class, data type 7 Clenshaw-Curtis quadrature 130, 196, 518, 519 Clenshaw’s recurrence formula 181ff., 196 for Chebyshev polynomials 193 stability 181ff. Clocking errors 899 cn function 269 Coarse-to-fine operator 873 Coarse-grid correction 873f. Coding arithmetic 910ff. checksums 896 decoding a Huffman-encoded message 905 Huffman 903ff. run-length 909 variable length code 903 Ziv-Lempel 903 see also Arithmetic coding; Huffman coding Coefficients binomial 215 for Gaussian quadrature 147ff. for Gaussian quadrature, nonclassical weight function 157ff., 797 for quadrature formulas 131ff., 797 Column degeneracy 32 Column operations on matrix 37, 40f. Column totals 630 Combinatorial minimization see Annealing Comite Consultatif International T ´ el´ egraphique ´ et Tel´ ephonique (CCITT) 897f., 909 ´ Communication theory, use in adaptive integration 727 Communications protocol 896 Comparison function for rejection method 290f. Complementary error function see Error function Complete elliptic integral see Elliptic integrals Complex arithmetic 23f., 176ff., 948ff. avoidance of in path integration 209 cubic equations 185 for linear equations 49f. quadratic equations 184 Complex error function 259 Complex plane fractal structure for Newton’s rule 367f. path integration for function evaluation 208ff., 271 poles in 111, 166, 208f., 213, 561, 573, 724f. Complex systems of linear equations 49f. complex.c utility functions 23f., 948ff. Compression of data 603, 889, 903ff., 910ff. Concordant pair for Kendall’s tau 642f. Condition number 61, 85 Confidence level 692f., 696ff. Confidence limits bootstrap method 692f. and chi-square 693f. confidence region, confidence interval 692 on estimated model parameters 689ff. by Monte Carlo simulation 689ff. from singular value decomposition (SVD) 698 Confluent hypergeometric function 210, 246 Conjugate directions 414f., 421ff. Conjugate gradient method biconjugate 84 compared to variable metric method 425f. elliptic partial differential equations 833 for minimization 396f., 420ff., 812f., 824 minimum residual method 85 preconditioner 85f. for sparse system 83ff., 606 and wavelets 606 Conservative differential equations 732f. Constrained linear inversion method 808ff. Constrained linear optimization see Linear programming Constrained optimization 394 Constraints, deterministic 813ff. Constraints, linear 431 Contingency coefficient C 631 Contingency table 628ff., 644 statistics based on chi-square 630ff. statistics based on entropy 632ff. continue construction 14 Continued fraction 169ff. Bessel functions 240f. convergence criterion 171 equivalence transformation 172 evaluation 169ff. evaluation along with normalization condition 247 even and odd parts 172, 217, 222 even part 255, 257 exponential integral 222 Fresnel integral 255 incomplete beta function 227 incomplete gamma function 217 Lentz’s method 171, 219 modified Lentz’s method 171 Pincherle’s theorem 181 ratio of Bessel functions 246 rational function approximation 170, 217, 227 recurrence for evaluating 170f. and recurrence relation 181 sine and cosine integrals 257 Steed’s method 170f. tangent function 169 typography for 169 Continuous variable (statistics) 628 Control structures 6, 8ff. bad 14
Index 969 Conventions in C programs 25ff. routine for 258f Convergence series 257 accelerated,for series 166ff Cosine transform see Fast Fourier transform of algorithm forπ915 (FFT);Fourier transform criteria for353,398f,410,489f.,495 Coulomb wave function 210,240 684f,767 Courant condition 838,841,843,845 eigenvalues accelerated by shifting 477f multidimensional 855 golden ratio 354,406 Courant-Friedrichs-Lewy stability criterion see of golden section search 398f. Courant condition of Levenberg-Marquardt method 684f Covariance linear 353,400 a priori 705 of QL method 477f. in general linear least squares 673,677 quadratic57,358,364f.,415f,427,915 matrix,by Cholesky decomposition 98, ate353,359f,364f. 673 recurrence relation 181 matrix,of errors 805,817 of Ridders'method 358 matrix,is inverse of Hessian matrix 685 series vs.continued fraction 169f. matrix,when it is meaningful 695f. and spectral radius 865ff.,871 in nonlinear models 685,687 convert_matrix()utility 945 relation to chi-square 695ff. Convex sets,use in inverse problems 813f. from singular value decomposition (SVD) Convolution 698 denoted by asterisk 498 in straight line fitting 663 finite impulse response(FIR)538 CR method see Cyclic reduction (CR) of functions 498.509 Cramer's V 631 of large data sets 543f. Crank-Nicolson method 848,853,855 for multiple precision arithmetic 918 CRC (cyclic redundancy check)896ff. multiplication as 918 CRC-12898 necessity for optimal filtering 542 CRC-16 polynomial 898 overlap-add method 544 CRC-CCITT 898 overlap-save method 543f. Creativity,essay on 8 and polynomial interpolation 120 Critical (Nyquist)sampling 500,550 relation to wavelet transform 1592 Cross(denotes matrix outer product)73 theorem 498,538ff.553 Crosstabulation analysis 629 theorem,discrete 538ff. see also Contingency table treatment of end effects 540 Crout's algorithm 44ff.,53 use of FFT 530,538ff. Cubic equations 183ff.,367 wraparound problem 540 Cubic spline interpolation 113ff. Cooley-Tukey FFT algorithm 509 see also Spline Co-processor,floating point 894 Cumulative binomial distribution 226,229 Copyright rules xvi Cumulative Poisson function 221 Cornwell-Evans algorithm 825 related to incomplete gamma function 221 Corporate promotion ladder 336f. Curvature matrix see Hessian matrix Corrected two-pass algorithm 613 cvector()utility 943 Correction,in multigrid method 872 Cycle,in multigrid method 874 Correlation coefficient (linear)636ff. Cyclic Jacobi method 466 Correlation function 498 Cyclic reduction (CR)857f.,861f. autocorrelation 498,546,565 Cyclic redundancy check(CRC)896ff. and Fourier transforms 498 Cyclic tridiagonal systems 74f. theorem 498.545 treatment of end effects 546 using FFT 545f. Danielson-Lanczos lemma 504f..532 Wiener-Khinchin theorem 498.574 Data Correlation,statistical 609f.,628 assigning keys to 897 Kendall's tau 640,642ff. continuous vs.binned 620 linear correlation coefficient 636ff.,664 entropy 632ff,903f. linear related to least square fitting 636 essay on 609 664 fitting 656ff. nonparametric or rank statistical 639ff. fraudulent 661 among parameters in a fit 663,673,676 glitches in 659 in random number generators 277 iid (independent and identically distributed) Spearman rank-order coefficient 640f. 691 sum squared difference of ranks 640 modeling 656ff. Cosine function,recurrence 178 serial port 899 Cosine integral 255,257ff. smoothing 610,650ff. continued fraction 257 statistical tests 609ff
Index 969 Conventions in C programs 25ff. Convergence accelerated, for series 166ff. of algorithm for π 915 criteria for 353, 398f., 410, 489f., 495, 684f., 767 eigenvalues accelerated by shifting 477f. golden ratio 354, 406 of golden section search 398f. of Levenberg-Marquardt method 684f. linear 353, 400 of QL method 477f. quadratic 57, 358, 364f., 415f., 427, 915 rate 353, 359f., 364f. recurrence relation 181 of Ridders’ method 358 series vs. continued fraction 169f. and spectral radius 865ff., 871 convert_matrix() utility 945 Convex sets, use in inverse problems 813f. Convolution denoted by asterisk 498 finite impulse response (FIR) 538 of functions 498, 509 of large data sets 543f. for multiple precision arithmetic 918 multiplication as 918 necessity for optimal filtering 542 overlap-add method 544 overlap-save method 543f. and polynomial interpolation 120 relation to wavelet transform 592 theorem 498, 538ff., 553 theorem, discrete 538ff. treatment of end effects 540 use of FFT 530, 538ff. wraparound problem 540 Cooley-Tukey FFT algorithm 509 Co-processor, floating point 894 Copyright rules xvi Cornwell-Evans algorithm 825 Corporate promotion ladder 336f. Corrected two-pass algorithm 613 Correction, in multigrid method 872 Correlation coefficient (linear) 636ff. Correlation function 498 autocorrelation 498, 546, 565 and Fourier transforms 498 theorem 498, 545 treatment of end effects 546 using FFT 545f. Wiener-Khinchin theorem 498, 574 Correlation, statistical 609f., 628 Kendall’s tau 640, 642ff. linear correlation coefficient 636ff., 664 linear related to least square fitting 636, 664 nonparametric or rank statistical 639ff. among parameters in a fit 663, 673, 676 in random number generators 277 Spearman rank-order coefficient 640f. sum squared difference of ranks 640 Cosine function, recurrence 178 Cosine integral 255, 257ff. continued fraction 257 routine for 258f. series 257 Cosine transform see Fast Fourier transform (FFT); Fourier transform Coulomb wave function 210, 240 Courant condition 838, 841, 843, 845 multidimensional 855 Courant-Friedrichs-Lewy stability criterion see Courant condition Covariance a priori 705 in general linear least squares 673, 677 matrix, by Cholesky decomposition 98, 673 matrix, of errors 805, 817 matrix, is inverse of Hessian matrix 685 matrix, when it is meaningful 695ff. in nonlinear models 685, 687 relation to chi-square 695ff. from singular value decomposition (SVD) 698 in straight line fitting 663 CR method see Cyclic reduction (CR) Cramer’s V 631 Crank-Nicolson method 848, 853, 855 CRC (cyclic redundancy check) 896ff. CRC-12 898 CRC-16 polynomial 898 CRC-CCITT 898 Creativity, essay on 8 Critical (Nyquist) sampling 500, 550 Cross (denotes matrix outer product) 73 Crosstabulation analysis 629 see also Contingency table Crout’s algorithm 44ff., 53 Cubic equations 183ff., 367 Cubic spline interpolation 113ff. see also Spline Cumulative binomial distribution 226, 229 Cumulative Poisson function 221 related to incomplete gamma function 221 Curvature matrix see Hessian matrix cvector() utility 943 Cycle, in multigrid method 874 Cyclic Jacobi method 466 Cyclic reduction (CR) 857f., 861f. Cyclic redundancy check (CRC) 896ff. Cyclic tridiagonal systems 74f. Danielson-Lanczos lemma 504f., 532 Data assigning keys to 897 continuous vs. binned 620 entropy 632ff., 903f. essay on 609 fitting 656ff. fraudulent 661 glitches in 659 iid (independent and identically distributed) 691 modeling 656ff. serial port 899 smoothing 610, 650ff. statistical tests 609ff